Elliptic Problems in with Critical and Singular Discontinuous Nonlinearities

Abstract

Let Ω be a bounded domain in R N , N ≥ 3 with smooth boundary, a > 0, λ > 0 and 0 < δ < 3 be real numbers. Define 2 * := 2N N − 2 and the characteristic function of a set A by χA. We consider the following critical problem with singular and discontinuous nonlinearity:We study the existence and the global multiplicity of solutions to the above problem.1991 Mathematics Subject Classification. 35J20, 35J60.

A Global multiplicity result for a very singular critical nonlocal equation

“…Using similar ideas as in (ZA) case, it can be shown that v is a weak solution of (P λ ). Then the rest of the proof follows exactly same as Lemma 3.3 of [9] or Lemma 2.7 of [13]. Proof of Theorem 1.3: The proof follows directly from Proposition 5.1 (Appendix) and Theorem 1.2 of [1].…”

Existence of positive solutions for a class of singular and quasilinear elliptic problems with critical exponential growth

“…A similar problem with the Laplacian operator in R 2 was studied by Saoudi and Kratou in [35]. In [16] Dhanya, Prashanth, Sreenadh and Tiwari considered the singular case with critical exponential growth and discontinous nonlinearity. The inhomogeneous singular Neumann case was studied in [36].…”

“…But the nonlinearities have polynomial growth. 2) In [16], [20], [35] and [36] were studied the singular case with a nonlinearities with exponential growth. However, here we study problems with a general operator which brings some technical difficulties.…”

Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities

“…If a = λ and b = 1, and q ∈ (0, 1), authors proved a global multiplicity result. While in [3,14], researchers improvised the results of [26] and proved the global multiplicity result for q ∈ (0, 3). In [28], Hirano, Saccon, and Shioji studied the problem (1.1) with a = λ and b = 1, and q ∈ (0, 1).…”